# On functional equations for the elliptic dilogarithm

## TL;DR

This paper investigates the structure of the pre-Bloch group for elliptic curves over algebraically closed fields, showing it can be generated by functions of degree up to three, and applies this to simplify relations involving the elliptic dilogarithm.

## Contribution

It proves that the pre-Bloch group of an elliptic curve's function field is generated by functions of degree at most three, simplifying the understanding of elliptic Bloch relations.

## Key findings

- Pre-Bloch group generated by degree ≤ 3 functions
- Elliptic Bloch relations reducible to antisymmetry and degree 3 relations
- Simplification of relations for the elliptic dilogarithm

## Abstract

Let $E$ be an elliptic curve over an algebraically closed field of characteristic 0. We prove that the pre-Bloch group of the function field of $E$ can be generated by the functions of degree not higher than 3. We apply this result to the elliptic dilogarithm function defined by S. Bloch. He has shown that any element of the pre-Bloch group gives a (so-called elliptic Bloch) relation between the values of the so-called Elliptic dilogarithm. We conclude that any elliptic Bloch relation can be reduced to the antisymmetry relation and the elliptic Bloch relations for the functions of degree 3.